Minimal Height Companion Matrices for Euclid Polynomials

Abstract

We define Euclid polynomials $E_{k+1}(\lambda ) = E_{k}(\lambda )\left( E_{k}(\lambda ) - 1\right) + 1$ and $E_{1}(\lambda ) = \lambda + 1$ in analogy to Euclid numbers $e_k = E_{k}(1)$. We show how to construct companion matrices $\mathbb {E}_k$, so $E_k(\lambda ) = {\text {det}}\left( \lambda \mathbf {I} - \mathbb {E}_{k}\right)$, of height 1 (and thus of minimal height over all integer companion matrices for $E_{k}(\lambda )$). We prove various properties of these objects, and give experimental confirmation of some unproved properties.